\newproblem{lay:1_1_13}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.1.13}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Clara Susana Rey Abad, Oct. 30, 2013} \\}{}

  % Problem statement
	
	Solve the equation system:
	\begin{center}
	 $\begin{array}{rcr}
	  x_1      -3x_3&=& 8\\
		2x_1+2x_2+9x_3&=& 7\\
		      x_2+5x_3&=&-2\\
	  \end{array}$
	\end{center}
	
}{
   % Solution
	Let us construct the augmented system matrix
	\begin{center}
		$\left(\begin{array}{rrr|r}
		   1 & 0 & -3 &  8 \\
			 2 & 2 &  9 &  7 \\
			 0 & 1 &  5 & -2
		\end{array}\right)$
	\end{center}
	
	Now, we apply row operations to solve it
	\begin{center}
		\begin{tabular}{cc}
			 $\mathbf{r}_2\leftrightarrow \mathbf{r}_3$ &
			 $\left(\begin{array}{rrr|r}
				 1 & 0 & -3 &  8\\
				 0 & 1 &  5 & -2 \\
				 2 & 2 &  9 &  7		
				 \end{array}\right)$ \\
			 $\mathbf{r}_3\leftarrow \mathbf{r}_3-2\mathbf{r}_1$ &
			 $\left(\begin{array}{rrr|r}
				 1 &  0  &  -3 &  8 \\
				 0 &  1  &   5 & -2 \\
				 0 &  2  &  15 & -9
			 \end{array}\right)$ \\
			
			$\mathbf{r}_3\leftarrow \mathbf {r}_3-2\mathbf{r}_2$ &
			$\left(\begin{array}{rrr|r}
			   1 &  0 &  -3  &  8 \\
				 0 &  1 &   5  & -2 \\
				 0 &  0 &   5  & -5
				\end{array}\right)$\\
				
			$\mathbf{r}_2\leftarrow \mathbf {r}_2-\mathbf{r}_3$ &
			$\left(\begin{array}{rrr|r}
			   1 &  0 &  -3  &  8 \\
				 0 &  1 &   0  &  3 \\
				 0 &  0 &   5  & -5
				\end{array}\right)$ \\
				
			\begin{tabular}{c}
			 $\mathbf{r}_3\leftarrow \mathbf {r}_3+\div 5$\\
			 $\mathbf{r}_1\leftarrow \mathbf {r}_1+3\mathbf{r}_3$\\
			\end{tabular}
			&
			$\left(\begin{array}{rrr|r}
			   1 &  0 &  0 &  5 \\
				 0 &  1 &  0 &  3  \\
				 0 &  0 &  1 & -1 
				\end{array}\right)$ \\
		\end{tabular}
	\end{center}
	
	Whe can deduce from the reduced echelon form that
	\begin{center}
	 $\begin{array}{rcr}
	  x_1 &=& 5\\
		x_2 &=& 3\\
		x_3 &=&-1\\
	  \end{array}$
	\end{center}
	
	 Therefore, there is a unique solution of the system. The equation system is compatible determinate.

}
\useproblem{lay:1_1_13}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
